On Solving Large-scale Limited-memory Quasi-newton Equations

We consider the problem of solving linear systems of equations with limited-memory members of the restricted Broyden class and symmetric rank-one matrices. In this paper, we present various methods for solving these linear systems, and propose a new approach based on a practical implementation of the compact representation for the inverse of these limited-memory matrices. Using the proposed approach has an additional benefit: The condition number of the system matrix can be computed efficiently. Numerical results suggest that the proposed method compares favorably in speed and accuracy to other algorithms and is competitive to methods available to only the Broyden-Fletcher-Goldfarb-Shanno update and the symmetric rank-one update. Limited-memory quasi-Newton methods, compact representation, restricted Broyden class of updates, symmetric rank-one update, Broyden-Fletcher-Goldfarb-Shanno update, Davidon-Fletcher-Powell update, Sherman-Morrison-Woodbury formula